In topology, a branch of mathematics, Quillen's Theorem A gives a sufficient condition for the classifying spaces of two categories to be homotopy equivalent. Quillen's Theorem B gives a sufficient condition for a square consisting of classifying spaces of categories to be homotopy Cartesian. The two theorems play central roles in Quillen's Q-construction in algebraic K-theory and are named after Daniel Quillen. The precise statements of the theorems are as follows.
In general, the homotopy fiber of B f : B C → B D {\displaystyle Bf:BC\to BD} is not naturally the classifying space of a category: there is no natural category F f {\displaystyle Ff} such that F B f = B F f {\displaystyle FBf=BFf} . Theorem B constructs F f {\displaystyle Ff} in a case when f {\displaystyle f} is especially nice.
References
Ara, Dimitri; Maltsiniotis, Georges (April 2018). "Un théorème A de Quillen pour les ∞-catégories strictes I : La preuve simpliciale". Advances in Mathematics. 328: 446–500. arXiv:1703.04689. doi:10.1016/j.aim.2018.01.018. Ara, Dimitri (October 2019). "A Quillen Theorem B for strict ∞-categories". Journal of the London Mathematical Society. 100 (2): 470–497. arXiv:1808.02650. doi:10.1112/jlms.12220. Ara, Dimitri; Maltsiniotis, Georges (2020). "Un théorème A de Quillen pour les ∞-catégories strictes II : la preuve ∞-catégorique". Higher Structures. 4 (1): 284–388. arXiv:1804.03241. doi:10.21136/HS.2020.07. Quillen, Daniel (1973), "Higher algebraic K-theory. I", Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math, vol. 341, Berlin, New York: Springer-Verlag, pp. 85–147, doi:10.1007/BFb0067053, ISBN 978-3-540-06434-3, MR 0338129 Srinivas, V. (2008), Algebraic K-theory, Modern Birkhäuser Classics (Paperback reprint of the 1996 2nd ed.), Boston, MA: Birkhäuser, ISBN 978-0-8176-4736-0, Zbl 1125.19300 Weibel, Charles (2013). The K-book: an introduction to algebraic K-theory. Graduate Studies in Math. Vol. 145. AMS. ISBN 978-0-8218-9132-2.
External links "geometric realization of categories". ncatlab.org. Lurie, Jacob. "7.2.3 Quillen's Theorem A for ∞-Categories". kerodon.net. Gurski, Nick; Johnson, Niles; Osorno, Angélica M. (2020). "2-categorical opfibrations, Quillen's Theorem B, and S − 1 S {\displaystyle S^{-1}S} ". arXiv:2010.11173 [math.CT].